The ring kinetics equations for hard spheres

The BBGKY hierarchy for hard spheres can be obtained by integration of the Eq. (2.62). Here we report the first two equations of the hierarchy derived in this way:

This set of equations is an open hierarchy which expresses the time evolution of the $s$-particle distribution function in terms of the $(s+1)$-th function.

Using again the Molecular Chaos assumption (Eq. (2.73)), the Boltzmann Equation (2.74) is immediately recovered from Eq. (2.94a).

Using the Enskog correction to the Molecular Chaos, Eq. (2.92), the Boltzmann-Enskog Equation is obtained.

As the density increases, the contributions of correlated collision sequences to the collision term become more and more important. At moderate densities, a simple way to take these correlations into account has been found in a cluster expansion of the $s-$particle distribution functions, defined recursively as

etc., where $g_2(1,2)=P_2(1,2)-P_1(1)P_1(2)$ accounts for pair correlations, $g_3(1,2,3)$ for triplet correlations, etc. The molecular chaos assumption implies $g_2(1,2)=0$. The basic assumption to obtain the ring kinetic equations is that the pair correlations are dominant and higher order ones can be neglected, i.e. $g_3=g_4=...=0$ in the above cluster expansion. The ring kinetic equations, obtained in this way, read:

with $\mathcal{X}(i,j)$ the operator that interchanges the particle labels $i$ and $j$. With further algebra and approximation one can derive the generalized Boltzmann equation in ring approximation. We do not give here this derivation, as it is not the aim of this work to review the entire ring kinetic theory in details, but just to give its basic ideas (which are the binary collision expansion Eqs. (2.56) and the cluster expansion, Eqs. (2.95)).

.